Exponents and the Laws of Exponents

Exponents

The Exponent of a number says how many times to use that number in a multiplication.
That is,
Exponents are the mathematical shorthand that tells us to multiply the same number by
itself for a specified number of times.

Example

The simpler way to write 5 * 5 * 5 * 5 * 5 * 5  as 5^6 or 56 .

Base : The number being multiplied by itself.
in our example, 5 the base .

Exponent : The number of times we are multiplying the base .
in our example , 6 is the exponent. Exponent are also called as power or index .  

Laws of Exponents 

1 . x ^ 1 = x  [ identity exponent ]

Example : 

• 5 ^ 1 = 5
• 8 ^ 1 = 8
• 0 ^ 1 = 0
• (-3) ^ 1 = -3 

2 . x ^ -n =( 1 / x)  ^ n [ Negative exponent ]

Note : 

x ^ -1 = 1 / x
( x / y ) ^ -n = ( y / x ) ^ n

Example

(5/3)^- 2 = (3/5)^2

                  = (3/5) * (3/5)
                  = (3*3) /( 5*5)
                  = 9 / 25 

3 . x ^ a * x ^ b = x ^ ( a + b ) [ product of powers ]

Example

3^2 * 3^3 = 3^(2+3)
  9   *  27    = 3^5
     243        = 243

4 . x ^ m / x ^ n = x ^ ( m - n ) [ Quotient of powers ]

Example

3 ^ 5 / 3 ^ 2 =  3 ^ ( 5 - 2 )
 243 /    9      =  3 ^ 3                    or         3*3*3*3*3 / 3*3 =  3 ^ 3
      27             =  27                        or                   3 * 3 * 3       =  27
                                                                                    27             =  27 

  5 . ( x ^ m ) ^ n = x ^ ( m * n )  [ power of power ]

Example

( 5 ^ 3 ) ^ 2  =  5 ^ ( 3 * 2 )
      125  ^ 2   =  5 ^ ( 6 )
       15,625    =  15,625

6 . ( x * y ) ^ m = (x ^ m) * (y ^ m) [ power of a product ]

Example

( 5 * 4 ) ^ 2  =  (5 ^ 2) * (4 ^ 2)
    20     ^ 2   =       25    *    16
         400      =           400

7 . ( x / y ) ^ m = x ^ m / y ^ m [ power of a quotient ]

Example

      (5/3)^2     =   (5^2) / (3^2)
(5/3) * (5/3)  =     25 / 9
    25 / 9          =    25 / 9

8 . x ^ 0 = 1   [ zero exponent ]

Note:
3/3 = 1
that is, 3^1/3^1 = 1  ( by law 4 )
                3^(1-1) = 1
                3^0 = 1

 
Example

•     5^0    =  1
• (1/5)^0 =  1

9 . x^(m/n) =  (x ^m)^1/n =             [Fractional exponent]

Example

5^(3/2) = (5^3) ^ (1/2)
               = 125 ^ (1/2)